Abstract

In this article, a new type of mappings that satisfies condition (B) is introduced. We study Pazy's type fixed point theorems, demiclosed principles, and ergodic theorem for mappings with condition (B). Next, we consider the weak convergence theorems for equilibrium problems and the fixed points of mappings with condition (B).

Keywords

1 Introduction

Let C be a nonempty closed convex subset of a real Hilbert space H. Let T : C → H be a mapping, and let F(T) denote the set of fixed points of T. A mapping T : C → H is said to be nonexpansive if ||Tx - Ty|| ≤ ||x - y|| for all x, y∈C. A mapping T : C → H is said to be quasi-nonexpansive mapping if F(T) ≠ ∅ and ||Tx - Ty|| ≤ ||x - y|| for all x∈C and y∈F(T).

In 2008, Kohsaka and Takahashi [1] introduced nonspreading mapping, and obtained a fixed point theorem for a single nonspreading mapping, and a common fixed point theorem for a commutative family of nonspreading mappings in Banach spaces. A mapping T : C → C is called nonspreading [1] if

2∥Tx-Ty∥2≤∥Tx-y∥2+∥Ty-x∥2

for all x, y∈C. Indeed, T : C → C is a nonspreading mapping if and only if

In 2011, Aoyama and Kohsaka [7] introduced α-nonexpansive mapping on Banach spaces. Let C be a nonempty closed convex subset of a Banach space E, and let α be a real number such that α < 1. A mapping T : C → E is said to be α-nonexpansive if

∥Tx-Ty∥2≤α∥Tx-y∥2+α∥Ty-x∥2+(1-2α)∥x-y∥2

for all x, y∈C.

Furthermore, we observed that Suzuki [8] introduced a new class of nonlinear mappings which satisfy condition (C) in Banach spaces. Let C be a nonempty subset of a Banach space E. Then, T : C → E is said to satisfy condition (C) if for all x, y∈C,

12∥x-Tx∥≤∥x-y∥⇒∥Tx-Ty∥≤∥x-y∥.

In fact, every nonexpansive mapping satisfies condition (C), but the converse may be false [8, Example 1]. Besides, if T : C → E satisfies condition (C) and F(T) ≠ ∅, then T is a quasi-nonexpansive mapping. However, the converse may be false [8, Example 2].

Similar to the above, we know that T is a TY-1 mapping, a TY-2 mapping, a hybrid mapping, (α, β)-generalized hybrid mapping, and T is a α-nonexpansive mapping.

On the other hand, the following iteration process is known as Mann's type iteration process [10] which is defined as

xn+1=αnxn+(1-αn)Txn,n∈ℕ,

where the initial guess x0 is taken in C arbitrarily and {αn } is a sequence in [0,1].

In 1974, Ishikawa [11] gave an iteration process which is defined recursively by

x1∈Cchosenarbitrary,yn:=(1-βn)xn+βnTxn,xn+1:=(1-αn)xn+αnTyn,

where {αn } and {βn } are sequences in [0,1].

In 1995, Liu [12] introduced the following modification of the iteration method and he called Ishikawa iteration method with errors: for a normed space E, and T : E → E a given mapping, the Ishikawa iteration method with errors is the following sequence

x1∈Echosenarbitrary,yn:=(1-βn)xn+βnTxn+un,xn+1:=(1-αn)xn+αnTyn+vn,

where {αn } and {βn } are sequences in [0,1], and {un } and {vn } are sequences in E with ∑n=1∞∥un∥<∞ and ∑n=1∞∥vn∥<∞.

In 1998, Xu [13] introduced an Ishikawa iteration method with errors which appears to be more satisfactory than the one introduced by Liu [12]. For a nonempty convex subset C of E and T : C → C a given mapping, the Ishikawa iteration method with errors is generated by

Furthermore, we observed that Phuengrattana [14] studied approximating fixed points of for a nonlinear mapping T with condition (C) by the Ishikawa iteration method on uniform convex Banach space with Opial property. Here, we also consider the Ishikawa iteration method for a mapping T with condition (C) and improve some conditions of Phuengrattana's result.

In this article, a new type of mappings that satisfies condition (B) is introduced. We study Pazy's type fixed point theorems, demiclosed principles, and ergodic theorem for mappings with condition (B). Next, we consider the weak convergence theorems for equilibrium problems and the fixed points of mappings with condition (B).

2 Preliminaries

Throughout this article, let ℕ be the set of positive integers and let ℝ be the set of real numbers. Let H be a (real) Hilbert space with inner product 〈·, ·〉 and norm || · ||, respectively. We denote the strongly convergence and the weak convergence of {xn } to x∈H by xn → x and xn⇀x, respectively. From [15], for each x, y∈H and λ∈ [0,1], we have

The following theorem shows that demiclosed principle is true for mappings with condition (B).

Theorem 3.1. Let C be a nonempty closed convex subset of a real Hilbert space H, and let T : C → C be a mapping with condition (B). Let {xn } be a sequence in C with xn⇀x andlimn→∞∥xn-Txn∥=0. Then Tx= x.

Proof By Remark 3.1, we get:

⟨Txn-Tx,x-Tx⟩≤⟨xn-x,Tx-x⟩+∥xn-Txn∥⋅∥x-Tx∥

for each n∈ℕ. By assumptions, 〈x - Tx , x - Tx 〉 ≤ 0. So, Tx = x.

Theorem 3.2. Let C be a nonempty closed convex subset of a real Hilbert space H, and let T : C → C be a mapping with condition (B). Then {Tnx} is a bounded sequence for some x∈C if and only if F(T) ≠ ∅.

Proof For each n∈ℕ, let xn := Tnx. Clearly, {xn } is a bounded sequence. By Lemma 2.1, there is a unique z∈C such that μn∥xn-z∥2=miny∈Cμn∥xn-y∥2. By Proposition 3.2, for each n∈ℕ,

(ii)⇒ (i): Take any x∈C and u∈F(T), and let x and u be fixed. Since T satisfies condition (B), ||Tnx - u|| ≤ ||Tn-1x - u|| for each n∈ℕ. Hence, limn→∞∥Tnx-u∥ exists and this implies that {Tnx} is a bounded sequence. By Lemma 2.4, there exists z∈F(T) such that limn→∞PF(T)Tnx=z. Clearly, z∈F(T). Besides, we have:

∥Snx-u∥≤1n∑k=0n-1∥Tkx-u∥≤∥x-u∥.

So, {Snx} is a bounded sequence. Then there exist a subsequence {Snix} of {Snx} and v∈C such that Snix⇀v. By the above proof, we have:

Assume that: {rn } ⊆ [a, ∞) for some a > 0 and lim infn→∞an(1-an)>0. Then xn⇀z, where z=limn→∞P(EP)∩F(T)xn.

Declarations

Competing interests

The authors declare that they have no competing interests, except Prof. L.-J. Lin was supported by the National Science Council of Republic of China while he worked on the publish, and C. S. Chuang was supported as postdoctor by the National Science Council of the Republic of China while he worked on this problem.

Authors' contributions

LJL is responsible for problem resign, coordinator, discussion, revise the important part, and submit. CSC is responsible for the important results of this article, discuss, and draft. ZTY is responsible for giving the examples of this types of problems, discussion. All authors read and approved the final manuscript.

Authors’ Affiliations

(1)

Department of Mathematics, National Changhua University of Education

(2)

Department of Electronic Engineering, Nan Kai University of Technology

Copyright

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.